Tutoring math 12 and some university courses, you get asked about counting. The tutor opens the discussion with a couple of examples.
Questions that ask, “How many ways can people be lined up for a photograph” or “how many five-card hands have two aces” or “how many ways can you fill out a multiple choice test” are all counting problems.
When an item, once used, cannot be used again, we call that without replacement. Counting without replacement can often be done using permute or choose, but there are other options. For a quick summary of permute and choose, see my article here.
When an item can be used over again, we call that with replacement. A question with replacement is often done using an exponential expression, as we shall see.
Example 1: How many ways can you arrange the letters of the word radio?
Solution: The answer is 5P5, or 120. You use P – which means Permute – because the order of the letters is what matters, and because the process is without replacement: that is, in a given arrangement, each letter can only appear once.
Example 2: How many five card hands from a standard deck of 52 cards have exactly 2 aces?
Solution: Here, the order doesn’t matter, but the process is still without replacement: if you get the queen of hearts, you cannot draw it again in the same hand. Without replacement, and where order doesn’t matter, points to Choose (aka Combination): Specifically, (4C2)(48C3) or 103776. The reasoning is as follows: from the 4 aces, choose 2: 4C2. Then, from the other 48 non-ace cards, choose 3: 48C3. Multiply the two results together since they both happen yet are independent of each other. Independent means that the choosing of the two aces has no influence on which non-ace cards will be chosen.
Example 3: How many possible ways can a 20-question multiple choice test be filled out, if each question has five options?
Solution: Order matters here, but it is with replacement: for instance, you can choose answer A over and over again. The number of ways you can fill out the test is 5^20. The reasoning is that you can fill out the first question five possible ways, then the second five possible ways, and so on. Your possibilities are (5)(5)(5)(5)……(5)=5^20. The number is absurdly large, so we’ll just refer to it as 5^20.
I’ll say more about counting in future posts. Hope your exams are going well:)
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.